Finite Energy Coordinates and Vector Analysis on Fractals

Hinz, Michael; Teplyaev, Alexander

doi:10.1007/978-3-319-18660-3_12

Cited by 12 publications

(18 citation statements)

References 51 publications

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“…Completing 1 a (X )/ ker • H with respect to • H we obtain a Hilbert space H, we refer to it as the space of generalized L 2 -vector fields associated with (E, F ). This is a version of a construction introduced in [17,18] and studied in [13,[49][50][51][52][53]55,77], see also the related sources [20,29,30,99]. The basic idea is much older, see for instance [15,Exercise 5.9], it dates back to ideas of Mokobodzki and LeJan.…”

Section: Resistance Forms and First Order Calculusmentioning

confidence: 99%

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Approximation of partial differential equations on compact resistance spaces

Hinz

Meinert

2021

Calc. Var.

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We consider linear partial differential equations on resistance spaces that are uniformly elliptic and parabolic in the sense of quadratic forms and involve abstract gradient and divergence terms. Our main interest is to provide graph and metric graph approximations for their unique solutions. For families of equations with different coefficients on a single compact resistance space we prove that solutions have accumulation points with respect to the uniform convergence in space, provided that the coefficients remain bounded. If in a sequence of equations the coefficients converge suitably, the solutions converge uniformly along a subsequence. For the special case of local resistance forms on finitely ramified sets we also consider sequences of resistance spaces approximating the finitely ramified set from within. Under suitable assumptions on the coefficients (extensions of) linearizations of the solutions of equations on the approximating spaces accumulate or even converge uniformly along a subsequence to the solution of the target equation on the finitely ramified set. The results cover discrete and metric graph approximations, and both are discussed.

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Section: Resistance Forms and First Order Calculusmentioning

confidence: 99%

“…A discussion of more general diffusion coefficients a should involve suitable coordinates, see [40,53,96]. In view of the fact that natural local energy forms on p.c.f.…”

Section: Remark 37mentioning

confidence: 99%

Approximation of partial differential equations on compact resistance spaces

Hinz

Meinert

2021

Calc. Var.

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“…There have been interests in the understanding of gradients of functions and non-linear partial differential equations on fractals with non-linearities involving first-order derivatives (see e.g. [16, 18–20, 33] and references therein). A new class of semi-linear parabolic equations involving singular measures on the Sierpinski gasket was proposed and studied in [27], where, among other things, a Feynman–Kac representation was obtained assuming the existence of weak solutions.…”

Section: Introductionmentioning

confidence: 99%

Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket

Li¹,

Qian

2019

Probab. Theory Relat. Fields

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By using analytic tools from stochastic analysis, we initiate a study of some non-linear parabolic equations on Sierpinski gasket, motivated by modellings of fluid flows along fractals (which can be considered as models of simplified rough porous media). Unlike the regular space case, such parabolic type equations involving non-linear convection terms must take a different form, due to the fact that convection terms must be singular to the “linear part” which defines the heat semigroup. In order to study these parabolic type equations, a new kind of Sobolev inequalities for the Dirichlet form on the gasket will be established. These Sobolev inequalities, which are interesting on their own and in contrast to the case of Euclidean spaces, involve two norms with respect to two mutually singular measures. By examining properties of singular convolutions of the associated heat semigroup, we derive the space-time regularity of solutions to these parabolic equations under a few technical conditions. The Burgers equations on the Sierpinski gasket are also studied, for which a maximum principle for solutions is derived using techniques from backward stochastic differential equations, and the existence, uniqueness, and regularity of its solutions are obtained.

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“…Our paper is a part of a broader program that aims to connect research on derivatives on fractals ( [8,9,[13][14][15][25][26][27][28]31,33,35,[37][38][39][40][41]46,51,54] and references therein) and on more general regular Dirichlet spaces [29,30,34] with classical and geometric analysis on metric measure spaces ( [6,10,12,[21][22][23][24][42][43][44][45]50] and references therein). In our previous article [36] we showed that on certain topologically one-dimensional spaces with a strongly local regular Dirichlet form one can prove a natural version of the Hodge theorem for 1-forms defined in 2 -sense: the set of harmonic 1-forms is dense in the orthogonal complement of the exact 1-forms.…”

Section: Introductionmentioning

confidence: 99%

“…In particular the reader can consult the papers [7,10,[42][43][44][45] and references therein. Our current paper is a step in a long-term program (see [29][30][31][32][33][34][35][36][37][38]) to develop parts of differential geometry and their applications to mathematical physics (see [3][4][5]) for spaces that carry diffusion processes but no other smooth structure. Our approach is somewhat complementary to the celebrated works [12,21,22] because, although our spaces are metrizable, we do not use any particular metric in an essential way, and we do not use functional inequalities.…”

Section: Introductionmentioning

confidence: 99%

Densely defined non‐closable curl on carpet‐like metric measure spaces

Hinz

Teplyaev

2018

Mathematische Nachrichten

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The paper deals with the possibly degenerate behaviour of the exterior derivative operator defined on 1‐forms on metric measure spaces. The main examples we consider are the non self‐similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one‐dimensional, they may have positive two‐dimensional Lebesgue measure and carry nontrivial 2‐forms. We prove that in this case the curl operator (and therefore also the exterior derivative on 1‐forms) is not closable, and that its adjoint operator has a trivial domain. We also formulate a similar more abstract result. It states that for spaces that are, in a certain way, structurally similar to Sierpinski carpets, the exterior derivative operator taking 1‐forms into 2‐forms cannot be closable if the martingale dimension is larger than one.

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Finite Energy Coordinates and Vector Analysis on Fractals

Cited by 12 publications

References 51 publications

Approximation of partial differential equations on compact resistance spaces

Approximation of partial differential equations on compact resistance spaces

Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket

Densely defined non‐closable curl on carpet‐like metric measure spaces

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